How many DIFFERENT quadrilaterals can be made by joining the dots on the 8-point circle?
What is the greatest number of counters you can place on the grid below without four of them lying at the corners of a square?
You have 4 red and 5 blue counters. How many ways can they be placed on a 3 by 3 grid so that all the rows columns and diagonals have an even number of red counters?
A dog is looking for a good place to bury his bone. Can you work out where he started and ended in each case? What possible routes could he have taken?
What is the smallest cuboid that you can put in this box so that you cannot fit another that's the same into it?
This challenge involves eight three-cube models made from interlocking cubes. Investigate different ways of putting the models together then compare your constructions.
Design an arrangement of display boards in the school hall which fits the requirements of different people.
Can you work out how many cubes were used to make this open box? What size of open box could you make if you had 112 cubes?
How will you go about finding all the jigsaw pieces that have one peg and one hole?
What is the best way to shunt these carriages so that each train can continue its journey?
Can you shunt the trucks so that the Cattle truck and the Sheep truck change places and the Engine is back on the main line?
Take a rectangle of paper and fold it in half, and half again, to make four smaller rectangles. How many different ways can you fold it up?
How can you arrange these 10 matches in four piles so that when you move one match from three of the piles into the fourth, you end up with the same arrangement?
This task, written for the National Young Mathematicians' Award 2016, involves open-topped boxes made with interlocking cubes. Explore the number of units of paint that are needed to cover the boxes. . . .
One face of a regular tetrahedron is painted blue and each of the remaining faces are painted using one of the colours red, green or yellow. How many different possibilities are there?
What shape has Harry drawn on this clock face? Can you find its area? What is the largest number of square tiles that could cover this area?
When I fold a 0-20 number line, I end up with 'stacks' of numbers on top of each other. These challenges involve varying the length of the number line and investigating the 'stack totals'.
Swap the stars with the moons, using only knights' moves (as on a chess board). What is the smallest number of moves possible?
What is the total area of the four outside triangles which are outlined in red in this arrangement of squares inside each other?
How many different ways can you find of fitting five hexagons together? How will you know you have found all the ways?
A tetromino is made up of four squares joined edge to edge. Can this tetromino, together with 15 copies of itself, be used to cover an eight by eight chessboard?
How many different triangles can you make on a circular pegboard that has nine pegs?
In how many ways can you fit two of these yellow triangles together? Can you predict the number of ways two blue triangles can be fitted together?
Cut four triangles from a square as shown in the picture. How many different shapes can you make by fitting the four triangles back together?
What happens to the area of a square if you double the length of the sides? Try the same thing with rectangles, diamonds and other shapes. How do the four smaller ones fit into the larger one?
Hover your mouse over the counters to see which ones will be removed. Click to remove them. The winner is the last one to remove a counter. How you can make sure you win?
10 space travellers are waiting to board their spaceships. There are two rows of seats in the waiting room. Using the rules, where are they all sitting? Can you find all the possible ways?
Players take it in turns to choose a dot on the grid. The winner is the first to have four dots that can be joined to form a square.
Here you see the front and back views of a dodecahedron. Each vertex has been numbered so that the numbers around each pentagonal face add up to 65. Can you find all the missing numbers?
Can you see why 2 by 2 could be 5? Can you predict what 2 by 10 will be?
A group activity using visualisation of squares and triangles.
Investigate the number of paths you can take from one vertex to another in these 3D shapes. Is it possible to take an odd number and an even number of paths to the same vertex?
Have a go at this 3D extension to the Pebbles problem.
Is it possible to rearrange the numbers 1,2......12 around a clock face in such a way that every two numbers in adjacent positions differ by any of 3, 4 or 5 hours?
Choose a box and work out the smallest rectangle of paper needed to wrap it so that it is completely covered.
Here are more buildings to picture in your mind's eye. Watch out - they become quite complicated!
What is the greatest number of squares you can make by overlapping three squares?
A game has a special dice with a colour spot on each face. These three pictures show different views of the same dice. What colour is opposite blue?
This 100 square jigsaw is written in code. It starts with 1 and ends with 100. Can you build it up?
Paint a stripe on a cardboard roll. Can you predict what will happen when it is rolled across a sheet of paper?
How many different cuboids can you make when you use four CDs or DVDs? How about using five, then six?
In each of the pictures the invitation is for you to: Count what you see. Identify how you think the pattern would continue.
Can you find ways of joining cubes together so that 28 faces are visible?
A toy has a regular tetrahedron, a cube and a base with triangular and square hollows. If you fit a shape into the correct hollow a bell rings. How many times does the bell ring in a complete game?
Can you work out what shape is made by folding in this way? Why not create some patterns using this shape but in different sizes?
A game for two players on a large squared space.
Have a look at what happens when you pull a reef knot and a granny knot tight. Which do you think is best for securing things together? Why?
Use the lines on this figure to show how the square can be divided into 2 halves, 3 thirds, 6 sixths and 9 ninths.
Imagine a 4 by 4 by 4 cube. If you and a friend drill holes in some of the small cubes in the ways described, how many will not have holes drilled through them?
A magician took a suit of thirteen cards and held them in his hand face down. Every card he revealed had the same value as the one he had just finished spelling. How did this work?